Page 254 - Bird R.B. Transport phenomena

238 Chapter 8 Polymeric Liquids
Upper plate oscillates with ^ T Fig- 8.2-2. Small-amplitude oscillatory
very small amplitude Ц ™ ~ ~ motion. For small plate spacing and
highly viscous fluids, the velocity pro-
(y, t) = y°y cos cot ^ e ш а У ^ e assume d to be linear.
v
x

Many ingenious devices have been developed to measure the three material func-
tions for steady shearing flow, and the theories needed for the use of the instruments are
explained in detail elsewhere. 2 See Problem 8C.1 for the use of the cone-and-plate instru-
ment for measuring the material functions.

Small-Amplitude Oscillatory Motion

A standard method for measuring the elastic response of a fluid is the small-amplitude
oscillatory shear experiment, depicted in Fig. 8.2-2. Here the top plate moves back and
forth in sinusoidal fashion, and with a tiny amplitude. If the plate spacing is extremely
small and the fluid has a very high viscosity, then the velocity profile will be nearly lin-
ear, so that v (y, t) = y°y cos (x)t, in which y°, a real quantity, gives the amplitude of the
x
shear rate excursion.
The shear stress required to maintain the oscillatory motion will also be periodic in
time and, in general, of the form
V = ~v'y° cos cot - r/'y 0 sin cot (8.2-4)

in which r\ and 77" are the components of the complex viscosity, 77* = 77' — z'77", which is a
function of the frequency. The first (in-phase) term is the "viscous response/' and the
second (out-of-phase) term is the "elastic response/' Polymer chemists use the curves of
г]'(to) and г)"(со) (or the storage and loss moduli, G' = rf'co and G" = rj'co) for "characteriz-
ing" polymers, since much is known about the connection between the shapes of these
curves and the chemical structure. For the Newtonian fluid, 77' = /x and 77" = 0.
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Steady-State Elongational Flow

A third experiment that can be performed involves the stretching of the fluid, in which
the velocity distribution is given by v = sz, v = —\ex, and v = -\ey (see Fig. 8.2-3),
z x y
where the positive quantity s is called the "elongation rate." Then the relation
_dv
r ~r = -77-^ (8.2-5)
xx
zz
defines the elongational viscosity rj, which depends on s. When s is negative, the flow is
referred to as biaxial stretching. For the Newtonian fluid it can be shown that 77 = 3/x, and
this is sometimes called the "Trouton viscosity."

Fig. 8.2-3. Steady elongational flow
v z = 62, v x = -jex, v y = -^ey with elongation rate s = dvjdz.

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J. D. Ferry, Viscoelastic Properties of Polymers, Wiley, New York, 3rd edition (1980).